Let us start from some historic note and references. The subject
was pioneered in the paper [1]}, where the calculations
for the two-parameters gauge were performed using Feynman
diagrams. Even more general diagrams-based calculation in the
non-minimal gauge were done in [2]}. The use of
the heat-kernel methods required the generalized Schwinger-DeWitt
technique [3]}, the results were applied to quantum GR
in [4]}. On the other hand, in [5]} it was
noted that one can simplify things by exploring the parametrization
ambiguity instead of the minimal gauge fixing.
The most general version of such a calculation [6]},
used the background field method with the parametrization
\(g_{\alpha \beta }&\,\,\longrightarrow \,\, & g^{\prime }_{\alpha \beta }= e^{2 \kappa r\sigma }\Big [g_{\alpha \beta }+ \kappa \big (\gamma _1\, \phi _{\alpha \beta }+ \gamma _2\, \phi \, g_{\alpha \beta } \big )\nonumber \\&&\,+\,\kappa ^2 \big (\gamma _3\, \phi _{\alpha \rho }\phi ^\rho _\beta + \gamma _4\, \phi _{\rho \omega } \phi ^{\rho \omega } \,g_{\alpha \beta }+ \gamma _5\,\phi \, \phi _{\alpha \beta }+ \gamma _6\, \phi ^2 \,g_{\alpha \beta }\big ) \Big ],\) 
